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Dr. Valerie R. Bencivenga

Economics

329

July 12

, 2018

MIDTERM EXAM #2

Instructions:

Answer the questions below in a blue exam book. There

are

9

questions worth

3

1

0

points. This

is a closed book exam. You may use a calculator (no device with wireless). The formula sheet

and probability

table are

at the end.

Show your work to receive credit.

The exam will last two hours.

Good luck!

(2

5

points)

1.

A small transportation company in the informal sector of Caracas moves construction materials. Below

is the probability dis

tribution of the number of jobs the company gets per day

(X)

:

X

1

2

3

4

P

X

(x)

0

.2

0

.3

0

.4

0

.1

a.

Give the expected number of jobs per day.

b.

Give the variance of the number of jobs per day.

c.

The company charges

$25

per job

.

It pays the driver

$20

per day

(regardless of the number of

jobs).

A

ssume for simplicity that the driver is the company’s only cost.

Write net revenue per day

(R)

as a linear transformation of jobs per day (X). (Net revenue is revenue minus costs.)

d.

What is expected

net revenue per day

?

e

.

What is the variance of net revenue per day?

(

15

points)

2

.

A bicycle shop has a sales business and a repair business. Let

X

represent net revenue

per month

from

the sales business. Expected net revenue

per month

from sales is

8

(

thousand

dollars)

and the variance

of net revenue

per month

from sales is

1.2

(thousand dollars

, squared)

. Let

Y

represent

net

revenue

per

month

from the repair business. Expected

net

revenue

per month

from repairs is

5

(

thousa

n

d dollars)

and the variance of

net

revenue

per month

from repairs is

0.64

(thousand dollars, squared).

The

covariance between

X

and

Y

is

0.2.

The shop has fixed costs every month equal to

9

(

thousand

dollars).

(Fixed costs are

rent, utilities,

insurance, wages

, etc., i.e. costs incurred regardless of how much net

revenue there is from the sales business and repair business.)

a.

Let

W

be profits

per month, in thousands of dollars

. Write

W

as a linear combination of

X

and

Y

.

b.

What are expecte

d monthly profits?

c.

What is the variance of monthly profits?

NAME: _____________________________________ EID: ___________________

You

must

return the exam questions along with your blue exam book, or you will

get a zero on the exam. Only your blue exam book will be graded.

 

(

40

points)

3

.

Here is the bivariate probability distribution of the prices of two stocks

.

X = price of stock #1

10

15

20

100

0

.1

0

.1

0

.1

Y = price of stock #2

200

0

.1

0

.2

0

.1

300

0

.1

0

.1

0

.1

a.

Calculate the expected price (mean) of

X,

and the expected price of

Y. Show your calculations.

(4 points)

b

.

Calculate

the covariance between

X

and

Y.

Show your calculation.

(10 points)

c

.

Are the prices of these two stocks independent?

Justify your answer

.

(6 points)

d

.

Give the conditional probability distribution of

Y

given that

the price of stock #1

(X)

is

20.

(4 points)

e

.

Compute

the expected price of stock #2

(mean of

Y

)

given

that

the price of stock #1

(X)

is

20

.

Are

the unconditional mean of

Y

from part a and the conditional mean of

Y

given

X = 20

the same?

(6 points)

f

.

Compute

the variance of

the

price of stock #2 (variance of

Y

).

(4 points)

g

.

Compute

the variance of the price of stock #2 (the variance of

Y

)

given that

the price of stock #1

(X)

is

20.

Are the unconditional variance of

Y

from part f and the conditional mean of

Y

given

X = 20

the same?

(6 points)

(

40

points)

4

.

A company submits

10

proposals. The probability any one proposal will lead to a contract

is

0

.

3.

The

outcomes of p

roposals are independen

t (“outcome

of a proposal

” means whether or not

the

proposal

leads to a contract)

.

a.

Let

X

represent the number of contracts the company obtains.

What is the expected number of

contracts the company will obtain from its

10

proposals?

What is t

he vari

ance of the number of

contracts?

(6 points)

b.

What is the probability that the company will

obtain at least

2

contracts?

(6 points)

c

.

The company has fixed costs of

400,000.

(Fixed costs do not depend on the number of

contracts or the number of proposals.)

E

ach proposal costs

100,000

to prepare.

Each

contract

generates

500,000

in net reven

ue.

Let

n

represent the number of proposals the company submits. Let

Y

represent the company’s

profits.

Write

p

rofit

(Y)

as a function of

the number of proposals

(n)

and

the number of

contracts

(X).

(10 points)

d.

If the company submits

10

proposals, calculate

expected profit and the variance of profit

.

(8 points)

e

.

What is the

smallest

number of proposals the company can submit yet still have positive

expected

profit?

(10 points)

(30 points)

5.

A company produces

sheets of Kevlar

fabric, for making bullet

-

proof vests. The company produces

10,000 square meters of fabric per day. The fabric is inspected for defects. Defects are randomly

located on the sheets of fabric. On average, there is

on

e

defect

per 100,000 square meters of f

abric.

a

.

What is the

expected number of

defects

in the

fabric produced

on

one day

?

b

.

What is the probability of

there being

at least one defect

in the

fabric produced

on

one day

?

c

.

What is the

expected number of

defects

in the

fabric produced

over a

three

-

day

period of time

?

d

.

What is the probability of finding 3

defects

in the

fabric produced

over a

three

-

day

period of

time

?

e

.

What is the probability of finding one

defect

in the

fabric produced

each day

, for

three days in a

row

?

f

.

Why are t

he

answers to parts d and e different from each other?

(

2

0 points)

6

.

Do

not

calculate your

answers.

A loan officer of a bank in a

village

in India

r

eceive

s

f

ifteen

loan

applications from farmers

,

and five

loan applications

from small businesses

.

The loan officer

has

funds to make nine loans.

T

he loan

off

i

cer chooses nine applic

ations randomly to receive the loans.

a.

W

hat is the probability that a majority of the loans are to small businesses?

b.

What is the probability that

a majority of the loans are to farmers?

c.

W

hat is the probability that

all of the loans are to

farmers?

d.

W

hat is the probability that

all of the loans are to small businesses?

(

5

0

points)

7

.

There are two phases of building a hotel. In phase one, the structure is constructed. In phase two, the

interior is finished (wallpaper, flooring, light fixtures, etc.).

Phase two f

ollows phase one.

Let

X

represent the number of months it takes phase one to be completed. Let

Y

represent the

number of months it takes phase two to be completed. Assume all months have the same number of

work days, and that it can take any fraction of one or more months to complete a phase.

X

is uniformly

distributed between

20

and

30,

and

Y

is u

niformly distributed between

2

0

and

30

.

X

and

Y

are

independent.

a.

State and graph the p.d.f. of

X.

State and graph the c.d.f. of

X.

(

16

points)

b.

Give the expected

value of

X.

Give the variance of

X.

(

6

points)

c.

State the joint p.d.f of

X

and

Y.

Briefly explain or show how you obtained your answer.

(

8

points)

d.

What is the probability that both

X

and

Y

will be greater than

2

5

?

(10 points)

e.

What is the probability that

the total number of months it takes to build the hotel will be

greater than

5

0

?

Let

T

represent the total number of months.

(10 points)

(

40

points)

8

.

For PhD students in economics, lengths of time it takes to complete their dissertations once they have

passed their qualifying exam (“time to completion”) are approximately normally distributed. Thus, “time

to completion” for a randomly

-

chosen economics PhD student is approximately a normal random

variable

(X).

Assume

X

~ N(31,7

2

)

where

X

is measured in months and a dissertation can take any

fraction of a month to complete.

a.

What is the probability that a randomly

-

chosen student will

take more than 28 months to

complete their dissertation

?

(5 points)

b

.

What number of months to comple

tion cuts off the 5% of students ta

king the longest to complete

their dissertations? F

ind the number of months

(which can be fractional)

such that 5% of

students tak

e longer than this to complete.

(5 points)

c

.

What is the probability that a student’s t

ime to completion will

be within one month of the

mean

?

(5 points)

d.

In a PhD

class, 25 students h

ave just passed their qualifying exam

. What is the probability that

the average

(mean)

time to completion in this

class

will be

more than 28 months? Assu

me that

these students are a random sample and that “time to completion” is independent across

students

.

(5 points)

e.

What is the probability that the average

(mean)

time to completion in this

class

of 25 students

will be within one month of the populat

ion mean?

(5 points)

f.

The probability you calculated in part d is larger than the probability you calculated in part a. The

probability you calculated in part e is larger than the probability you calculated in part c. Briefly

explain

why

.

(5 points)

g

.

How large

would a class

have to be to ensure that the probability is

at least 0

.9 that the average

time to completion is within one month of the population mean?

(10 points)

Keep going! Question 9 is on the next page.

(

50

points)

9

.

You

plan to

open a small business

in a resort town

. You are deciding whether to

produce

fudge

or

taffy

or

some of each

.

Let

X

represent

t

he number of

pounds of fudge

you

could

make and

sell per week

,

if you

put all of your

time into producing

fudge

.

Assume

X ~ N(

40,

16

).

Let

Y

represent t

he number of

pounds of taffy

you

could

make and

sell per week

,

if you

put all of your

time into producing

taff

y

.

Assume

Y ~ N(

6

0, 25).

The covariance between

X

and

Y

is

negative.

XY

10

  

(

squared pounds

).

You estimate that net revenue (selling price minus cost of production)

would be

12

dollars

per

pound

of

fudge, and

8

dollars

per

pound

of taffy

.

a.

Use

RX

to represent net revenue of your b

usiness

if you

produce

only

fudge

.

Write

RX

as a

function of

X.

What is expected net revenue?

What is the variance of net revenue?

(

6

points)

b.

What is the probability that

RX

would be

less than

450?

Greater than

510?

(

10

points)

c

.

Use

R

Y

to represent net revenue of your business

if you

produce

only

taff

y

.

Write

RY

as a

function of

Y.

What is expected net revenue? What is the variance of net revenue?

(

6

points)

d.

What is the probability that

RY

would be less than

450?

Greater than

510?

(

10

points)

You could divide your time between producing fudge

and producing taffy. If you spend half of your time

producing fudge, you can make and sell half as much (compared to producing only fudge), and similarly

for taffy.

e

.

Use

RD

to represent net

revenue of your business if you put

half

of your time into

pro

ducing

fudge

, and

half

of your time into

producing

taff

y

.

Write

RD

as a function of

X

and

Y.

What is

expected net revenue? What is the variance of net revenue?

(

6

points)

f

.

What is

the probability that

RD

would

be less than

450?

Greater than

510?

(

10

points)

g.

Which business plan (“only fudge”, “only taffy”, or “half of your time in each”) should you adopt if

your objective is to choose the one with the highest probability of net revenue greater than

510?

No additional calculations are necessary.

(1 point)

Which business play should you adopt if your objective is to choose the one with the smallest

probability of net revenue less than

450? No additional calculations are necessary. (1 point)